Rooted-tree solutions for tree games

Béal, Sylvain; Rémila, Éric; Solal, Philippe

doi:10.1016/j.ejor.2009.07.023

Cited by 34 publications

(22 citation statements)

References 8 publications

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“…The notion of hierarchical outcome has been introduced by Demange (2004) in the context of games on cycle-free graph structures. In this same context Herings, van der Laan and Talman (2008) propose the average of all hierarchical outcomes as the so-called average tree solution, whereas Béal, Rémila and Solal (2010) generalize this solution to the class of all weighted averages of the hierarchical outcomes. In this section we examine the class of weighted hierarchical solutions from the perspective of the river game without referring to the underlying graph-theoretical concepts.…”

Section: Weighted Hierarchical Solutionsmentioning

confidence: 99%

Fair Agreements for Sharing International Rivers with Multiple Springs and Externalities

Brink

Laan

Moes

2010

AIP Conference Proceedings

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In this paper we consider the problem of sharing water from a river among a group of agents (countries, cities, firms) located along the river. The benefit of each agent depends on the amount of water consumed by the agent. An allocation of the water among the agents is efficient when it maximizes the total benefits. To sustain an efficient water allocation, the agents can compensate each other by paying monetary transfers. Every water allocation and transfer schedule yields a welfare distribution, where the utility of an agent is equal to its benefit from the water consumption plus its monetary transfer (which can be negative). The problem of finding a fair welfare distribution can be modelled by a cooperative game.For a river with one spring and increasing benefit functions, Ambec and Sprumont (2002) propose the downstream incremental solution as the unique welfare distribution that is core-stable and satisfies the condition that no agent gets a utility payoff above its aspiration level. Ambec and Ehlers (2008) generalized the Ambec and Sprumont river game to river situations with satiable agents, i.e., the benefit function is decreasing beyond some satiation point. In such situations externalities appear, yielding a cooperative game in partition function form. In this paper we consider river situations with satiable agents and with multiple springs. For this type of river systems we propose the class of so-called weighted hierarchical solutions as the class of solutions satisfying several principles to be taken into account for solving water disputes. When every agent has an increasing benefit function (no externalities) then every weighted hierarchical solution is core-stable. In case of satiation points, it appears that every weighted hierarchical solution is independent of the externalities.

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Section: Weighted Hierarchical Solutionsmentioning

confidence: 99%

Fair Agreements for Sharing International Rivers with Multiple Springs and Externalities

Brink

Laan

Moes

2010

AIP Conference Proceedings

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“…Our characterization of the AT solution does not rely on the classical axioms of efficiency, contrary to all previous characterizations in the literature, and linearity, contrary to the characterizations in Mishra and Talman (2010) and Béal et al (2010) where the graph is also fixed. It is based on two axioms of invariance plus a punctual axiom.…”

Section: Introductionmentioning

confidence: 45%

“…A leaf of L is an agent in N who is incident to only one link. Following Béal et al (2010), the set of cones of L consists of N , ∅ and, for each {i, j} ∈ L, of the two connected components that are obtained after deletion of link {i, j}. Every cone except N is called a proper cone.…”

Section: Preliminariesmentioning

confidence: 99%

“…Other characterizations on the class of cycle-free graph games have been provided by van den Brink (2009), Mishra and Talman (2010), Béal et al (2010Béal et al ( , 2012b and Ju and Park (2012). Generalizations of the AT solution to the class of all graph games have been examined by Herings et al (2010) and Baron et al (2011).…”

Section: Introductionmentioning

confidence: 99%

“…The axiom of Invariance to irrelevant coalitions is based on the set of cones of a tree, which contains the grand coalition, the empty coalition, and every pair of components obtained by removing a link from the original tree. Only coalitions in this set are involved in the computation of the AT solution, and, more generally, in the computations of marginalist tree solutions in the sense of Béal et al (2010). The axiom of Invariance to irrelevant coalitions requires that the solution should prescribe the same payoff vector in two tree games where the worths of all cones are the same.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of the Average Tree solution and its kernel

Béal

Rémila²,

Solal³

2015

Journal of Mathematical Economics

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In this article, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We first derive direct-sum decompositions of the space of TU-games on a fixed tree, and two new basis for these spaces of TU-games. We then focus our attention on the Average (rooted)-Tree solution (see Herings, P., van der Laan, G., Talman, D., 2008. The Average Tree Solution for Cycle-free Games. Games and Economic Behavior 62, 77-92). We provide a basis for its kernel and a new axiomatic characterization by using the classical axiom for inessential games, and two new axioms of invariance, namely Invariance with respect to irrelevant coalitions and Weighted addition invariance on bi-partitions.

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